Numerical optimization (Electrical Machine) Part 1


Shape optimization for small DC motor

The application of the methods used is demonstrated by the optimisation of a small DC motor. For the optimisation a (4/4, 12) evolution strategy combined with the simulated annealing algorithm is used.

The objective is to minimise the overall material expenditure, determined by permanent magnet-, copper- and iron volume subject to a given torque of the example motor.


The use of penalty term in the form:


allows the evaluation of the objective function even if the torque constraint is violated.

The torque is computed by integrating the Maxwell stress tensor in the air gap region. Flux density dependent rotor iron losses were taken into account at a rated speed of 200 rpm and subtracted from the air gap torque to form the resulting output torque.

The overall dimensions and the slot geometry of the DC motor are described by 15 free design parameters. The free parameters are the n/2 edges of the polygon describing the rotor slot contour and the outer dimensions of rotor and stator as indicated in Fig. 10.42. The motor consists of a stator back iron with a 2-pole- Ferrite permanent magnet system and a rotor with six slots.

Geometrical definitions and design variables of the DC motor.

Fig, 10.42. Geometrical definitions and design variables of the DC motor.

The necessary two-dimensional field computation to evaluate the quality function, to compute the torque of the machine, is performed by standard two-dimensional finite element analysis. To ensure controlled accuracy, adaptive mesh generation is applied until a given error bound is fulfilled.

An initial mesh is generated from any geometry represented by non-overlapping polygons. Fig. 10.43 shows an initial and adaptive generated mesh for the example DC motor.

a) initial- and b) adaptively refined mesh.

Fig. 10.43. a) initial- and b) adaptively refined mesh.

Constraints result from fabrication conditions. The change of shape from a sub-optimal initial geometry to the final shape of the motor can be taken from Fig. 10.44. It can be noticed that the iron parts of the initial geometry are over dimensioned. The actual torque of this configuration was approximately 25% lower than the desired valuetmpF-731The optimised motor holds the torque recommended, which is achieved mainly by enlarging the winding copper volume by about 20%. The most significant change from start to final geometry can be seen in the halving of the iron volume. Consequently the iron parts are highly saturated, especially the teeth regions. In comparison to this, a test optimisation with neglected rotor iron loss results in a 10% smaller rotor diameter. Unfortunately, the permanent magnet material is brittle, which limits the minimum magnet height. The magnet volume decreases slightly. Along optimisation the overall volume off the motor was reduced by 38%. The rate of convergence is plotted in Fig. 10.45.

Motor shapes during optimisation (iteration step: quality).

Fig. 10.44. Motor shapes during optimisation (iteration step: quality).

Quality versus iteration counts.

Fig. 10.45. Quality versus iteration counts.

Pole shape optimisation of a synchronous generator

The aim is to optimise the pole shape of a 3-phase synchronous generator. The quality function is evaluated by a simplified analytical approach. At no load of the generator typically the generated voltage is desired to be sinusoidal. The time dependent sinusoidal output voltage requires a position dependent sinusoidal distributed flux distribution in the air gap. With DC field exciting current and a concentrated shaded pole, the sinusoidal field excitation is reached by influencing the air gap reluctance. The air gap lengthtmpF-735is a function of the circumferential angletmpF-736

Due to symmetry only half a pole pitch is used for the evaluation of the air gap flux density distribution. The following assumptions were made:

• the stator of the machine is spotless

• saturation of ferromagnetic parts is neglected

• inside the iron parts it should betmpF-737i.e. flux lines are perpendicular at the iron boundaries

• the flux lines are approximated by circular arcs

• pole flux leakage is neglected

• flux density in the interpolar gap is not present.

Geometry and co-ordinate system.

Fig. 10.46. Geometry and co-ordinate system.

Fig. 10.46 shows the geometry and co-ordinate system of the used configuration. The pole shape of half a pole pitch is approximated with n parts of a polygon. Between the sample of the polygon, linear interpolation of the flux density is applied. The objective variables are the y-co-ordinatestmpF-742of the samples of the polygon. The air gap length

tmpF-743restricts the optimisation problem.


With the objective function:


tmpF-748denotes the fundamental of the flux density distribution and the tmpF-749are the harmonics. Determination of the harmonics is performed by a fast Fourier transformation (FFT). With Ampere’s law on the path of integration as indicated in Fig. 10.46 the flux density of the position of interest is evaluated with:


Fig. 10.47 shows the initial and optimised pole shape. The variations of the pole contour for temporary iteration steps can be taken out from Fig. 10.48.

Pole shape and flux density; a) rectangle pole and b) optimised pole shape.

Fig. 10.47. Pole shape and flux density; a) rectangle pole and b) optimised pole shape.

Pole shape and qualityof iteration step k.

Fig. 10.48. Pole shape and qualitytmpF-757of iteration step k.

Table 10.4. Quality function during optimisation.


tmp11-760 tmp11-761


tmp11-762 tmp11-763


tmp11-764 tmp11-765


tmp11-766 tmp11-767


tmp11-768 tmp11-769


tmp11-770 tmp11-771


tmp11-772 tmp11-773


tmp11-774 tmp11-775